Tautological stable pair invariants of Calabi-Yau 4-folds

TL;DR

This paper introduces tautological stable pair invariants for Calabi-Yau 4-folds, conjectures their generating series in terms of Gopakumar-Vafa invariants, and verifies the conjecture in specific examples including the local resolved conifold.

Contribution

It proposes a new conjectural formula relating stable pair invariants on Calabi-Yau 4-folds to Gopakumar-Vafa invariants and verifies it in key examples.

Findings

Conjectural generating series formula involving Gopakumar-Vafa invariants.

Verification of the formula in several examples including the local resolved conifold.

Complete determination of invariants in the JS chamber confirming previous conjectures.

Abstract

Let $X$ be a Calabi-Yau 4-fold and $D$ a smooth divisor on it. We consider tautological complex associated with $L=\mathcal{O}_X(D)$ on the moduli space of Le Potier stable pairs and define its counting invariant by integrating the Euler class against the virtual class. We conjecture a formula for their generating series expressed using genus zero Gopakumar-Vafa invariants of $D$ and genus one Gopakumar-Vafa type invariants of $X$, which we verify in several examples. When $X$ is the local resolved conifold, our conjecture reproduces a conjectural formula of Cao-Kool-Monavari in the PT chamber. In the JS chamber, we completely determine the invariants and confirm one of our previous conjectures.